Problem: Simplify and expand the following expression: $ \dfrac{3t - 7}{5t + 3}+\dfrac{5t - 9}{2t - 2} $
Solution: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(5t + 3)(2t - 2)$ Multiply the first term by $\dfrac{2t - 2}{2t - 2}$ $ \begin{align*} \dfrac{3t - 7}{5t + 3} \times \dfrac{2t - 2}{2t - 2} & = \dfrac{(3t - 7)(2t - 2)}{(5t + 3)(2t - 2)} \\ & = \dfrac{6t^2 - 20t + 14}{(5t + 3)(2t - 2)}\end{align*} $ Multiply the second term by $\dfrac{5t + 3}{5t + 3}$ $ \begin{align*} \dfrac{5t - 9}{2t - 2} \times \dfrac{5t + 3}{5t + 3} & = \dfrac{(5t - 9)(5t + 3)}{(2t - 2)(5t + 3)} \\ & = \dfrac{25t^2 - 30t - 27}{(2t - 2)(5t + 3)}\end{align*} $ Now we have: $ = \dfrac{6t^2 - 20t + 14}{(5t + 3)(2t - 2)} + \dfrac{25t^2 - 30t - 27}{(2t - 2)(5t + 3)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{6t^2 - 20t + 14 + 25t^2 - 30t - 27}{(5t + 3)(2t - 2)} $ $ = \dfrac{31t^2 - 50t - 13}{(5t + 3)(2t - 2)}$ Expand the denominator: $ = \dfrac{31t^2 - 50t - 13}{10t^2 - 4t - 6}$